Smarandache curves according to Sabban frame of fixed pole curve belonging to the Bertrand curves pair Süleyman Şenyurt, Yasin Altun, and Ceyda Cevahir Citation: AIP Conference Proceedings 1726, 020045 (2016); doi: 10.1063/1.4945871 View online: http://dx.doi.org/10.1063/1.4945871 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1726?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On characterizations of some special curves of timelike curves according to the Bishop frame of type-2 in Minkowski 3-space AIP Conf. Proc. 1726, 020082 (2016); 10.1063/1.4945908 A variational characterization and geometric integration for Bertrand curves J. Math. Phys. 54, 043508 (2013); 10.1063/1.4800767 Design of the pole pieces of an electromagnet according to the Garber–Henry–Hoeve model Rev. Sci. Instrum. 56, 771 (1985); 10.1063/1.1138172 The shapes of pair polarizability curves J. Chem. Phys. 72, 298 (1980); 10.1063/1.438847 Investigating the physical nature of the Coriolis effects in the fixed frame Am. J. Phys. 45, 631 (1977); 10.1119/1.10780
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Smarandache Curves According to Sabban Frame of Fixed Pole Curve Belonging to the Bertrand Curves Pair S¨uleyman S¸enyurt1,a) , Yasin Altun1,b) and Ceyda Cevahir1,c) 1
Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey. a)
Corresponding author: senyurtsuleyman@hotmail.com b) yasinaltun2852@gmail.com c) Ceydacevahir@gmail.com
Abstract. In this paper, we investigate the Smarandache curves according to Sabban frame of fixed pole curve which drawn by the unit Darboux vector of the Bertrand partner curve. Some results have been obtained. These results were expressed as the depends Bertrand curve.
Mathematics Subject Classification (2010). 53A04. Keywords. Bertrand curves pair, Fixed pole curve, Smarandache curves, Sabban frame, Geodesic curvature.
INTRODUCTION AND PRELIMINARIES A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve [1]. K. Tas¸k¨opr¨u, M. Tosun studied special Smarandache curves according to Sabban frame on S 2 [2]. S¸enyurt and C ¸ alıs¸kan investigated special Smarandache curves in terms of Sabban frame for fixed pole curve and spherical indicatrix and they gave some characterization of Smarandache curves [4, 6]. Let α : I → E 3 be a unit speed curve denote by {T, N, B} the moving Frenet frame. For an arbitrary curve α ∈ E 3 , with first and second curvature, κ and τ respectively, the Frenet formulae is given by [7, 8] T 0 = κN,
N 0 = −κT + τB,
B0 = −τN
(1)
the vector W is called Darboux vector defined by W = τT + κB. If we consider the normalization of the Darboux C = and [5]
1 kWk W
we have, sin ϕ =
τ kWk
and cos ϕ =
C = sin ϕT + cos ϕB
κ kWk
(2)
where ∠(W, B) = ϕ. Theorem 1 Let α : I → E 3 and α∗ : I → E 3 be the C 2 -class differentiable unit speed two curves and the amounts of {T (s), N(s), B(s), κ(s), τ(s)} and {T ∗ (s), N ∗ (s), B∗ (s), κ∗ (s), τ∗ (s)} are entirely Frenet- serret aparataus of the curves α and the Bertrand partner α∗ , respectively, then T∗
=
cos θT − sin θB, N ∗ = N, B∗ = sin θT + cos θB, κ∗ =
λκ − sin2 θ ∗ sin2 θ , τ = 2 λ(1 − λκ) λτ
(3) (4)
where ∠(T, T ∗ ) = θ, [8].
International Conference on Advances in Natural and Applied Sciences AIP Conf. Proc. 1726, 020045-1–020045-4; doi: 10.1063/1.4945871 Published by AIP Publishing. 978-0-7354-1373-3/$30.00
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Theorem 2
Let (α, α∗ ) be a Bertrand curves pair in E3 . We have between unit Darboux vectors [3], C ∗ = −C.
(5)
Theorem 3 Let γ be a unit speed spherical curve. We denote s as the arc-length parameter of γ . Let us denote t(s) = γ0 (s), and we call t(s) a unit tangent vector of γ. We now set a vector d(s) = γ(s) ∧ t(s) along γ. This frame is called the Sabban frame of γ on S 2 (Sphere of unit radius). Then we have the following spherical Frenet formulae of γ, [2, 6] γ0
=
t, t0 = −γ + κg d, d0 = −κg t
(6)
where κg is called the geodesic curvature of γ on S 2 and κg = ht0 , di.
(7)
SMARANDACHE CURVES ACCORDING TO SABBAN FRAME OF FIXED POLE CURVE BELONGING TO THE BERTRAND CURVES PAIR In this section, we investigate Smarandache curves according to the Sabban frame of fixed pole (C ∗ ). Let αC∗ (s) = C ∗ be a unit speed regular spherical curves on S 2 . We denote sC ∗ as the arc-lenght parameter of fixed pole (C ∗ ) αC∗ (s) = C ∗ (s). Differentiating (8), we have
(8)
TC ∗ = cos ϕ∗ T ∗ − sin ϕ∗ B∗
and
C ∗ ∧ TC ∗ = N ∗ .
From the equation C∗
= sin ϕ∗ T ∗ + cos ϕ∗ B∗ , TC ∗ = cos ϕ∗ T ∗ − sin ϕ∗ B∗ , C ∗ ∧ TC ∗ = N ∗
(9)
is called the Sabban frame of fixed pole curve (C ∗ ). From the (6) κg = hTC ∗ 0 , C ∗ ∧ TC ∗ i =⇒ κg =
kW ∗ k . ϕ∗0
Then from the (4) we have the following spherical Frenet formulae of (C ∗ ): C ∗0
=
TC ∗ ,
TC ∗ 0 +
kW ∗ k kW ∗ k ∗ C ∧ TC ∗ , (C ∗ ∧ TC ∗ )0 = − ∗0 TC ∗ . ∗0 ϕ ϕ
(10)
i.) C ∗ TC ∗ -Smarandache Curves Let S 2 be a unit sphere in E 3 and suppose that the unit speed regular Bertrand partner curve αC∗ (s) = C ∗ (s) lying fully on S 2 . In this case, C ∗ TC ∗ - Smarandache curve can be defined by 1 β1 (s) = √ (C ∗ + TC ∗ ). 2
(11)
Substituting the equation (9) into equation (11), we reach 1 β1 (s) = √ sin ϕ∗ + cos ϕ∗ T ∗ + cos ϕ∗ − sin ϕ∗ B∗ . 2
(12)
Differentiating (11), we can write the tangent vector of β1 -Smarandache curve according to Bertrand partner curve (ϕ∗0 − sin ϕ∗ ) kW ∗ k (ϕ∗0 + sin ϕ∗ ) ∗ T∗ + q N∗ − q B. T β1 = q 2 2 2 ∗0 ∗ 2 ∗0 ∗ 2 ∗0 ∗ 2 2ϕ + kW k 2ϕ + kW k 2ϕ + kW k
(13)
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Differentiating (13), we get √ √ √ ϕ∗0 4 2(χ1 sin ϕ∗ + χ2 cos ϕ∗ ) ∗ χ3 ϕ∗0 4 2 ϕ∗0 4 2(χ1 cos ϕ∗ − χ2 sin ϕ∗ ) ∗ 0 ∗ T β1 = T + B 2 2 N + 2 kW ∗ k2 + ϕ∗0 2 kW ∗ k2 + ϕ∗0 2 kW ∗ k2 + ϕ∗0 2
(14)
where χ1
=
χ3
=
kW ∗ k 0 kW ∗ k kW ∗ k kW ∗ k 4 kW ∗ k 0 kW ∗ k kW ∗ k 2 + , χ2 = −2 − 3 ∗0 2 − − ∗0 ∗0 ∗0 ∗0 ϕ ϕ ϕ ϕ ϕ ϕ∗0 ϕ∗0 ∗ ∗ ∗ kW k kW k 3 kW k 0 2 ∗0 + + . ∗0 ϕ ϕ ϕ∗0 −2 −
(15)
Considering the equations (12) and (13), it easily seen that (C ∗ ∧ TC ∗ )β1
=
kW ∗ k(cos ϕ∗ + sin ϕ∗ ) ∗ ϕ∗0 kW ∗ k(cos ϕ∗ + sin ϕ∗ ) ∗ T − q N∗ + q B. q 2kW ∗ k2 + 4ϕ∗0 2 2kW ∗ k2 + 4ϕ∗0 2 2kW ∗ k2 + 4ϕ∗0 2
(16)
Substituting the (3) and (4) into equation (12), (13), (14) and (16), Sabban aparataus of the β1 -Smarandache curve according to Bertrand curve kWk ϕ0 (cos ϕ + sin ϕ) ϕ0 (sin ϕ − cos ϕ) 1 T− p N+ p B, β1 (s) = √ (− sin ϕ−cos ϕ)T +(− cos ϕ+sin ϕ)B , T β1 = p 2 2ϕ02 + kWk2 2ϕ02 + kWk2 2ϕ02 + kWk2 (C ∗ ∧ TC ∗ )β1
ϕ0 kWk(cos ϕ + sin ϕ) kWk(cos ϕ − sin ϕ) T− p N− p B, p 2 02 2 02 2kWk + 4ϕ 2kWk + 4ϕ 2kWk2 + 4ϕ02
√ √ √ χ3 ϕ04 2 ϕ04 2(−χ1 sin ϕ − χ2 cos ϕ) ϕ04 2(−χ1 cos ϕ + χ2 sin ϕ) T− B, N+ kWk2 + 2ϕ02 2 kWk2 + 2ϕ02 2 kWk2 + 2ϕ02 2
=
T β0 1
=
where χ1
=
χ3
=
kWk 0 kWk kWk 4 kWk 0 kWk kWk 2 kWk + , χ2 = −2 − 3 0 2 − − 0 0 0 0 ϕ ϕ ϕ ϕ ϕ ϕ0 ϕ0 kWk 3 kWk kWk + 2 0 0. 2 0 + ϕ ϕ0 ϕ −2 −
(17)
Geodesic curvatures of the β1 (sβ1 )-Smarandache curve according to Bertrand partner and Bertrand curves, recpectively, κgβ1
=
kW ∗ k
1 2+
kW ∗ k 2 25 ) ϕ∗0
ϕ∗0
χ1 −
kWk kW ∗ k 1 kWk χ2 + 2χ3 , κgβ1 = χ1 − 0 χ2 + 2χ3 . 5 ∗0 0 2 kWk ϕ ϕ 2 + ϕ0 ) 2 ϕ
ii.) TC ∗ (C ∗ ∧ TC ∗ )-Smarandache Curves TC ∗ (C ∗ ∧ TC ∗ )-Smarandache curve can be defined by 1 β2 (s) = √ (TC ∗ + C ∗ ∧ TC ∗ ). 2
(18)
Solving the above equation by substitution of TC ∗ and C ∗ ∧ TC ∗ from (9), (3) and (4), we reach β2 -Smarandache curve according to Bertrand partner and Bertrand curves, respectively, β2 (s) =
1 1 √ cos ϕ∗ T ∗ + N ∗ − sin ϕ∗ B∗ , β2 (s) = √ − cos ϕT + N + sin ϕB . 2 2
(19)
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Geodesic curvature of the β2 (sβ2 )-Smarandache curve according to Bertrand curve κgβ2 =
kWk 2 0 δ1 − δ2 + δ3 5 ϕ kWk 2 2
1 1+2
ϕ0
where δ1 =
kWk kWk 3 kWk kWk kWk kWk kWk kWk 0 kWk kWk 0 + +2 0 0 , δ2 = −1−3 0 2 −2 0 4 − , δ3 = − 0 2 −2 0 4 + . 0 0 0 0 ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ0
iii.) C ∗ TC ∗ (C ∗ ∧ TC ∗ )-Smarandache Curves C ∗ TC ∗ (C ∗ ∧ TC ∗ )-Smarandache curve can be defined by 1 β3 (s) = √ (C ∗ + TC ∗ + C ∗ ∧ TC ∗ ). 3
(20)
Solving the above equation by substitution of C ∗ , TC ∗ and C ∗ ∧ TC ∗ from (9), (3) and (4), we reach β3 -Smarandache curve according to Bertrand partner and Bertrand curves, respectively, 1 1 β3 (s) = √ (sin ϕ∗ + cos ϕ∗ )T ∗ + N ∗ + (cos ϕ∗ − sin ϕ∗ )B∗ , β3 (s) = √ (− sin ϕ − cos ϕ)T + N + (sin ϕ − cos ϕ)B . 3 3 Geodesic curvature of the β3 (sβ3 )-Smarandache curve according to Bertrand curve, κgβ3
ϕ05 2kWk − ϕ0 ρ1 − ϕ05 kWk + ϕ0 ρ2 + ϕ05 2ϕ0 − kWk ρ3 = 25 √ 4 2 ϕ02 − ϕ0 kWk + kWk2
where ρ1 ρ2 ρ3
kWk kWk 2 kWk kWk 0 kWk kWk +4 0 − +2 0 3+ 2 0 −1 , 0 0 0 ϕ ϕ ϕ ϕ ϕ ϕ kWk kWk 2 kWk 3 kWk 4 kWk 0 kWk = −2 + 2 0 − 4 0 + −2 0 − 1+ 0 , ϕ ϕ ϕ0 ϕ ϕ0 ϕ kWk kWk kWk kWk 0 kWk kWk 2− 0 . = 2 0 −4 0 2+4 0 3−2 0 4+ ϕ ϕ ϕ ϕ ϕ0 ϕ = −2 + 4
ACKNOWLEDGEMENT This work was supported by BAP (The Scientific Research Projects Coordination Unit), Ordu University.
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